Narrow Results

Inspired by Wilson's paper on sectional curvatures of Kähler moduli, we consider a natural Riemannian metric on a hypersurface {f=1} in a real vector space, defined using the Hessian of a homogeneous polynomial f. We give examples to answer a question posed by Wilson about when this metric has nonpositive curvature. Also, we exhibit a large class of polynomials f on R³ such that the associated metric has constant negative curvature. We ask if our examples, together with one example by Dubrovin, are the only ones with constant negative curvature. This question can be rephrased as an appealing question in classical invariant theory, involving the "Clebsch covariant". We give a positive answer for polynomials of degree at most 4, as well as a partial result in any degree.

Let G be a reductive affine algebraic group and let X be an affine algebraic G-variety. We establish a (poly)stability criterion for points x ∈ X in terms of intrinsically defined closed subgroups H_x of G and relate it with the numerical criterion of Mumford and with Richardson and Bate–Martin–Röhrle criteria, in the case X = G^N. Our criterion builds on a close analogue of a theorem of Mundet and Schmitt on polystability and allows the generalization to the algebraic group setting of results of Johnson–Millson and Sikora about complex representation varieties of finitely presented groups. By well established results, it also provides a restatement of the non-abelian Hodge theorem in terms of stability notions.

We present a generalized version of classical geometric invariant theory à la Mumford where we consider an affine algebraic group G acting on a specific affine algebraic variety X. We define the notions of linearly reductive and of unipotent action in terms of the G fixed point functor in the category of (G, 𝕜[X])-modules. In the case that X = {⋆} we recuperate the concept of linearly reductive and of unipotent group. We prove in our "relative" context some of the classical results of GIT such as: existence of quotients, finite generation of invariants, Kostant–Rosenlicht's theorem and Matsushima's criterion. We also present a partial description of the geometry of such linearly reductive actions.

We consider a finite permutation group acting naturally on a vector space $V$ over a field $𝕜$. A well-known theorem of Göbel asserts that the corresponding ring of invariants $𝕜{[V]}^G$ is generated by the invariants of degree at most $\left(\genfrac{}{}{0.0pt}{}{dimV}{2}\right)$. In this paper, we show that if the characteristic of $𝕜$ is zero, then the top degree of vector coinvariants $𝕜{[V^m]}_G$ is also bounded above by $\left(\genfrac{}{}{0.0pt}{}{dimV}{2}\right)$, which implies the degree bound $\left(\genfrac{}{}{0.0pt}{}{dimV}{2}\right)+1$ for the ring of vector invariants $𝕜{[V^m]}^G$. So, Göbel’s bound almost holds for vector invariants in characteristic zero as well.

This note proves that rings of vector invariants of permutation groups acting on an $n$-dimensional vector space in characteristic zero are generated by its invariants of degree less than or equal to $max\{n,\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)\}$. This improves our previous result [F. Reimers and M. Sezer, Vector invariants of permutation groups in characteristic zero, Internat. J. Math.35(3) (2024) 2350111] and fully extends Göbel’s theorem [M. Göbel, Computing bases for rings of permutation-invariant polynomials, J. Symbolic Comput.19(4) (1995) 285–291] to the case of vector invariants.

The aspects of many particle systems, as far as their entanglement is concerned, is highlighted. To this end we briefly review the bipartite measures of entanglement and the entanglement of pairs both for systems of distinguishable and indistinguishable particles. The analysis of these quantities in macroscopic systems shows that close to quantum phase transitions, the entanglement of many particles typically dominates that of pairs. This leads to an analysis of a method to construct many-body entanglement measures. SL-invariant measures are a generalization to quantities as the concurrence, and can be obtained with a formalism containing two (actually three) orthogonal antilinear operators. The main drawback of this antilinear framework, namely to measure these quantities in the experiment, is resolved by a formula linking the antilinear formalism to an equivalent linear framework.

By using of the invariant theory, we have studied the generalized time-dependent giant spin model. The dynamical and geometric phases are given, respectively. The Aharonov–Anandan phase is also obtained under the cyclical evolution.

Many-body Hilbert space is a functional vector space with the natural structure of an algebra, in which vector multiplication is ordinary multiplication of wave functions. This algebra is finite-dimensional, with exactly $N{!}^{d-1}$ generators for N identical particles, bosons or fermions, in d dimensions. The generators are called shapes. Each shape is a possible many-body vacuum. Shapes are natural generalizations of the ground-state Slater determinant to more than one dimension. Physical states, including the ground state, are superpositions of shapes with symmetric-function coefficients, for both bosons and fermions. These symmetric functions may be interpreted as bosonic excitations of the shapes. The algebraic structure of Hilbert space described here provides qualitative insights into long-standing issues of many-body physics, including the fermion sign problem and the microscopic origin of bands in the spectra of finite systems.

Quantum dynamical properties of a general time-dependent coupled oscillator are investigated based on the theory of two-dimensional (2D) dynamical invariants. The quantum dynamical invariant of the system satisfies the Liouville–von Neumann equation and it coincides with its classical counterpart. The mathematical formula of this invariant involves a cross term which couples the two oscillators mutually. However, we show that, by introducing two pairs of annihilation and creation operators, it is possible to uncouple the original invariant operator so that it becomes the one that describes two independent subsystems. The eigenvalue problem of this decoupled quantum invariant can be solved by using a unitary transformation approach. Through this procedure, we eventually obtain the eigenfunctions of the invariant operator and the wave functions of the system in the Fock state. The wave functions that we have developed are necessary in studying the basic quantum characteristics of the system. In order to show the validity of our theory, we apply our consequences to the derivation of the fluctuations of canonical variables and the uncertainty products for a particular 2D oscillatory system whose masses are exponentially increasing.

By using the Lewis–Riesenfeld invariants theory, we investigate the quantum dynamics of a two-dimensional (2D) time-dependent coupled oscillator. We introduce a unitary transformation and show the conditions under which the invariant operator is uncoupled to describe two simple harmonic oscillators with time-independent frequencies and unit masses. The decouplement allows us to easily obtain the corresponding eigenstates. The generalization to three-dimensional (3D) time-dependent coupled oscillator is briefly discussed where a diagonalized invariant, which is exactly the sum of three simple harmonic oscillators, is obtained under specific conditions on the parameters.

We find explicitly the multiplicities in the (mixed) trace cocharacter sequence of two 3 × 3 matrices over a field of characteristic 0 and show that asymptotically they behave as polynomials of seventh degree. As a consequence we obtain also the multiplicities of certain irreducible characters in the cocharacter sequence of the polynomial identities of 3 × 3 matrices.

For the standard $27$-dimensional representation $V$ of the exceptional group $G$ of type $E_6$ we prove that $(SL(V),G)$ is a Donkin pair if and only if the characteristic of a ground field is greater than $13$. We also develop an elementary approach to describe submodule structure of any exterior power of $V$.

We consider the algebra of invariants of $d$-tuples of $n\times n$ matrices under the action of the orthogonal group by simultaneous conjugation over an infinite field of characteristic $p$ different from two. It is well known that this algebra is generated by the coefficients of the characteristic polynomial of all products of generic and transpose generic $n\times n$ matrices. We establish that in case $0<p\le n$ the maximal degree of indecomposable invariants tends to infinity as $d$ tends to infinity. In other words, there does not exist a constant $C(n)$ such that it only depends on $n$ and the considered algebra of invariants is generated by elements of degree less than $C(n)$ for any $d$. This result is well-known in case of the action of the general linear group. On the other hand, for the rest of $p$ the given phenomenon does not hold. We investigate the same problem for the cases of symmetric and skew-symmetric matrices.

The algebra of ${\mathrm{\text{GL}}}_n$-invariants of m-tuples of $n\times n$ matrices with respect to the action by simultaneous conjugation is a classical topic in case of infinite base field. On the other hand, in case of a finite field generators of polynomial invariants are not known even for a pair of $2\times 2$ matrices. Working over an arbitrary field we classified all ${\mathrm{\text{GL}}}_2$-orbits on m-tuples of $2\times 2$ nilpotent matrices for all $m>0$. As a consequence, we obtained a minimal separating set for the algebra of ${\mathrm{\text{GL}}}_2$-invariant polynomial functions of m-tuples of $2\times 2$ nilpotent matrices. We also described the least possible number of elements of a separating set for an algebra of invariant polynomial functions over a finite field.

For a finite-dimensional representation $V$ of a group $G$ over a field $F$, the degree of reductivity $\delta (G,V)$ is the smallest degree $d$ such that every nonzero fixed point $v\in V^G\setminus \left\{0\right\}$ can be separated from zero by a homogeneous invariant of degree at most $d$. We compute $\delta (G,V)$ explicitly for several classes of modular groups and representations. We also demonstrate that the maximal size of a cyclic subgroup is a sharp lower bound for this number in the case of modular abelian $p$-groups.

For each simple Lie algebra $𝔤$ (excluding, for trivial reasons, type $C$), we find the lowest possible degree of an invariant second-order PDE over the adjoint variety in $\mathbb{P}𝔤$, a homogeneous contact manifold. Here a PDE $F(x^i,u,u_i,u_{ij})=0$ has degree $\le d$ if $F$ is a polynomial of degree $\le d$ in the minors of $(u_{ij})$, with coefficient functions of the contact coordinate $x^i$, $u$, $u_i$ (e.g., Monge–Ampère equations have degree 1). For $𝔤$ of type $A$ or $G_2$, we show that this gives all invariant second-order PDEs. For $𝔤$ of types $B$ and $D$, we provide an explicit formula for the lowest-degree invariant second-order PDEs. For $𝔤$ of types $E$ and $F_4$, we prove uniqueness of the lowest-degree invariant second-order PDE; we also conjecture that uniqueness holds in type $D$.

Notions of rank abound in the literature on tensor decomposition. We prove that strength, recently introduced for homogeneous polynomials by Ananyan–Hochster in their proof of Stillman’s conjecture and generalized here to other tensors, is universal among these ranks in the following sense: any non-trivial Zariski-closed condition on tensors that is functorial in the underlying vector space implies bounded strength. This generalizes a theorem by Derksen–Eggermont–Snowden on cubic polynomials, as well as a theorem by Kazhdan–Ziegler which says that a polynomial all of whose directional derivatives have bounded strength must itself have bounded strength.

To compute the unique formal normal form of families of vector fields with nilpotent linear part, we choose a basis of the Lie algebra consisting of orbits under the action of the nilpotent linear part. This creates a new problem: to find explicit formulas for the structure constants in this new basis. These are well known in the 2D case, and recently expressions were found for the 3D case by ad hoc methods. The goal of the this paper is to formulate a systematic approach to this calculation. We propose to do this using a rational method for the inversion of the Clebsch–Gordan coefficients. We illustrate the method on a family of 3D vector fields and compute the unique formal normal form for the Euler family both in the 2D and 3D cases, followed by an application to the computation of the unique normal form of the Rössler equation.

Let $M=G/H$ be an $(n+1)$-dimensional homogeneous manifold and $J^k(n,M)=:J^k$ be the manifold of $k$-jets of hypersurfaces of $M$. The Lie group $G$ acts naturally on each $J^k$. A $G$-invariant partial differential equation of order $k$ for hypersurfaces of $M$ (i.e., with $n$ independent variables and $1$ dependent one) is defined as a $G$-invariant hypersurface ${ℰ}\subset J^k$. We describe a general method for constructing such invariant partial differential equations for $k\ge 2$. The problem reduces to the description of hypersurfaces, in a certain vector space, which are invariant with respect to the linear action of the stability subgroup $H^{(k-1)}$ of the $(k-1)$-prolonged action of $G$. We apply this approach to describe invariant partial differential equations for hypersurfaces in the Euclidean space ${𝔼}^{n+1}$ and in the conformal space ${𝕊}^{n+1}$. Our method works under some mild assumptions on the action of $G$, namely: A1) the group $G$ must have an open orbit in $J^{k-1}$, and A2) the stabilizer $H^{(k-1)}\subset G$ of the fiber $J^k\to J^{k-1}$ must factorize via the group of translations of the fiber itself.

This is the first in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field $K$, we construct a standard monomial basis for the arc space of the determinantal variety over $K$. As an application, we prove the arc space analogue of the first and second fundamental theorems of invariant theory for the general linear group.